Two-Lines: A Valid Alternative to the Invalid Testing of U-Shaped Relationships with Quadratic Regressions
36 Pages Posted: 22 Aug 2017 Last revised: 21 Oct 2017
Date Written: October 18, 2017
Many psychological theories predict u-shaped relationships: the effect of x is positive for low values of x, but negative for high values, or vice-versa. These predictions are tested primarily via quadratic regressions; here I show such approach is essentially never valid. I propose instead estimating a regression with two separate lines, one for ‘low’ and one for ‘high’ values of x. A u-shape is present if the two slopes have opposite sign and are individually significant. A procedure to set the breakpoint for ‘low’ vs ‘high’ is proposed, the Robin Hood algorithm, and shown to be superior to all alternatives considered, including maximizing fit. The problems with the quadratic and advantages of the two-lines are demonstrated via simulations and examples from published psychological research.
Keywords: regression, interactoin, quadratic, u-shape, specification error
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